Longitudinal focusing is of great importance in space-charge-dominated bunched beams. The bunches lengthen rapidly due to the strong space-charge forces during transport. The UMD electron ring is designed to operate with bunches having rectangular and parabolic profiles. Three induction cavities along the ring will be employed for longitudinal focusing. In this paper, we first briefly review the longitudinal dynamics of parabolic and rectangular bunches. The design of the induction cavities with their high repetition rate modulators is then presented. The preliminary test results of lower voltage and repetition rate version of two different modulators and the SPICE simulation results show that modulators work well.
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